Notes and Links
All times are best-case and approximate.
Train times: http://www.nationalrail.co.uk/
Tube times: every few minutes, but subject to delays.
See map of stations and color-coded lines.
Service tends to be worse on Sundays / after 21:00 and is sometimes cancelled with little warning.
Train and Tube tickets can be bought from appropriate windows/machines.
Train tickets are good on any train that day (or within 30 days if
"open").
Tube tickets are sometimes integrated with train, and can be
single-use, all-day-anywhere, etc. See Oyster below.

Bus times: http://www.nationalexpress.com/
Other (limited) bus services exist around London and between airports.
Bus tickets are easily bought online here and printed. Tip:
getting an earlier bus is fine (though could fill up, in theory),
but being late costs money, so I regularly buy the time-slot 1 or 2
bus-departures
later than my flight would usually dictate.
Plan to arrive 2 hrs pre-flight and leave about 1hr post-flight for
passport check and baggage claim.

Heathrow and Gatwick are the main airports, with regular and some
discount airlines. Tickets to/from are all the
same,
but note that Heathrow has Terminals 1-4 and Gatwick has a North and a
South
Terminal. Transfer within one airport just costs time.
Discount airlines like RyanAir,
EasyJet, and a few
others have their own online booking, and land in or
near (careful!) many European cities. For example, see
destinations/airlines for Stansted
Airport here.
Other airports (which terminal?) with airlines vs. destinations:Heathrow
(or here),
Gatwick
(or here),
Luton
(or here
or here),
and London
City.

Oyster and saving money:
To save some money on the Tube, get a magnetic Oyster card at a
London Underground ticket office;
the Oyster card will cost you 3GBP (refundable deposit) plus however
much virtual-money you load onto it
for Pay As You Go. For example, if I were coming to visit London and
Cambridge for a few days,
(I'd either take the bus to Cambridge and then travel from there or...)
I'd land at Heathrow, buy an Oyster card with 15 GBP on it, ride the
Picadilly line to King's Cross, buy
a one-way (single) National Rail ticket to Cambridge from the machines
at the top of the stairs (on your left),
then make day-trips to London, and finally, turn in the Oyster card at
Heathrow for the deposit and any unused value.
From Cambridge (or to Cambridge), the single-day round-trip (return)
rail tickets are the cheapest option, since
"open" return tickets cost 2x the one-way price.